# ISU Logic Seminar: Closed groups generated by generic measure preserving transformations

**Author: Lona**

**Author: Lona**

Speaker: Slawomir Solecki (Cornell)

Abstract:

The behavior of a measure preserving transformation, even a generic one, is highly non-uniform. In contrast to this observation, a different picture of a very uniform behavior of the closed group generated by a generic measure preserving transformation $T$ has emerged. This picture included substantial evidence that pointed to these groups (for a generic $T$) being all topologically isomorphic to a single group, namely, $L_{0}$—the topological group of all Lebesgue measurable functions from $[0,1]$ to the circle. In fact, Glasner and Weiss asked if this is the case.

We will describe the background touched on above (including all the relevant definitions) and outline a proof of the following theorem that answers the Glasner–Weiss question in the negative: for a generic measure preserving transformation $T$, the closed group generated by $T$ is **not** topologically isomorphic to $L_{0}$. The proof rests on an analysis of unitary representations of $L_{0}$.